@IEEEtranBSTCTL{IEEEexample:BSTcontrol,
  CTLdash_repeated_names = "no"
}


@article{schmid2010jfm,
  title={Dynamic mode decomposition of numerical and experimental data},
  author={Schmid, Peter J},
  journal={Journal of fluid mechanics},
  volume={656},
  pages={5--28},
  year={2010},
  doi={10.1017/S0022112010001217},
  publisher={Cambridge University Press}
}
@book{Kutz2016book,
  title={Dynamic mode decomposition: data-driven modeling of complex systems},
  author={Kutz, J Nathan and Brunton, Steven L and Brunton, Bingni W and Proctor, Joshua L},
  year={2016},
  doi={10.1137/1.9781611974508},
  publisher={SIAM}
}
@software{Falcon_PyTorch_Lightning_2019,
author = {Falcon, William and {The PyTorch Lightning team}},
doi = {10.5281/zenodo.3828935},
license = {Apache-2.0},
month = mar,
title = {{PyTorch Lightning}},
url = {https://github.com/Lightning-AI/lightning},
version = {1.4},
year = {2019}
}
@article{kaptanoglu2022pysindy,
  doi = {10.21105/joss.03994},
  url = {https://doi.org/10.21105/joss.03994},
  year = {2022},
  publisher = {The Open Journal},
  volume = {7},
  number = {69},
  pages = {3994},
  author = {Alan A. Kaptanoglu and Brian M. de Silva and Urban Fasel and Kadierdan Kaheman and Andy J. Goldschmidt and Jared Callaham and Charles B. Delahunt and Zachary G. Nicolaou and Kathleen Champion and Jean-Christophe Loiseau and J. Nathan Kutz and Steven L. Brunton},
  title = {PySINDy: A comprehensive {P}ython package for robust sparse system identification},
  journal = {Journal of Open Source Software}
  }
@article{pedregosa2011scikit,
  title={Scikit-learn: Machine learning in {P}ython},
  author={Pedregosa, Fabian and Varoquaux, Ga{\"e}l and Gramfort, Alexandre and Michel, Vincent and Thirion, Bertrand and Grisel, Olivier and Blondel, Mathieu and Prettenhofer, Peter and Weiss, Ron and Dubourg, Vincent and others},
  journal={the Journal of machine Learning research},
  volume={12},
  pages={2825--2830},
  year={2011},
  publisher={JMLR. org}
}

@article{schmid2022dynamic,
	author = {Schmid, Peter J},
	journal = {Annual Review of Fluid Mechanics},
	pages = {225--254},
	title = {Dynamic mode decomposition and its variants},
	volume = {54},
  doi={10.1146/annurev-fluid-030121-015835},
	year = {2022}
}


@article{ljung2010arc,
  title={Perspectives on system identification},
  author={Ljung, Lennart},
  journal={Annual Reviews in Control},
  volume={34},
  number={1},
  pages={1--12},
  year={2010},
  doi={10.1016/j.arcontrol.2009.12.001},
  publisher={Elsevier}
}
@article{wright1999numerical,
  title={Numerical optimization},
  author={Wright, Stephen and Nocedal, Jorge and others},
  journal={Springer Science},
  volume={35},
  number={67-68},
  pages={7},
  doi={10.1007/978-0-387-40065-5},
  year={1999}
}
@article{paszke2019pytorch,
  title={Pytorch: An imperative style, high-performance deep learning library},
  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}
@article{demo2018pydmd,
  title={PyDMD: Python dynamic mode decomposition},
  author={Demo, Nicola and Tezzele, Marco and Rozza, Gianluigi},
  journal={Journal of Open Source Software},
  volume={3},
  number={22},
  pages={530},
  doi={10.21105/joss.00530},
  year={2018}
}

@book{Brunton2019book,
  title={Data-driven science and engineering: Machine learning, dynamical systems, and control},
  author={{Brunton}, Steven L. and Kutz, J Nathan},
  year={2022},
  doi={10.1017/9781108380690},
  publisher={Cambridge University Press}
}
@article{scikit-learn,
 title={Scikit-learn: Machine Learning in {P}ython},
 author={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V.
         and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P.
         and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and
         Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.},
 journal={Journal of Machine Learning Research},
 volume={12},
 pages={2825--2830},
 year={2011}
}
@article{brunton2022modern,
author = {Brunton, Steven L. and Budi\v{s}i\'{c}, Marko and Kaiser, Eurika and Kutz, J Nathan},
title = {Modern {K}oopman Theory for Dynamical Systems},
journal = {SIAM Review},
volume = {64},
number = {2},
pages = {229-340},
year = {2022},
doi = {10.1137/21M1401243}
}

@article{Budivsic2012chaos,
  title={Applied {K}oopmanism},
  author={Budi{\v{s}}i{\'c}, Marko and Mohr, Ryan and Mezi{\'c}, Igor},
  journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
  volume={22},
  number={4},
  pages={047510},
  year={2012},
  doi={10.1063/1.4772195},
  publisher={American Institute of Physics}
}
@article{klus2017data,
  title={Data-driven model reduction and transfer operator approximation},
  author={Klus, Stefan and N{\"u}ske, Feliks and Koltai, P{\'e}ter and Wu, Hao and Kevrekidis, Ioannis and Sch{\"u}tte, Christof and No{\'e}, Frank},
  journal={Journal of Nonlinear Science},
  volume={28},
  number={3},
  pages={985--1010},
  year={2018},
  doi={10.1007/s00332-017-9437-7},
  publisher={Springer}
}

@article{Mezic2013arfm,
  title={Analysis of fluid flows via spectral properties of the {K}oopman operator},
  author={Mezi{\'c}, Igor},
  journal={Annual Review of Fluid Mechanics},
  volume={45},
  pages={357--378},
  year={2013},
  doi={10.1146/annurev-fluid-011212-140652},
  publisher={Annual Reviews}
}
@article{Li2017chaos,
  title={Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the {K}oopman operator},
  author={Li, Qianxiao and Dietrich, Felix and Bollt, Erik M and Kevrekidis, Ioannis G},
  journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
  volume={27},
  number={10},
  pages={103--111},
  doi={10.1063/1.4993854},
  year={2017},
  publisher={AIP Publishing LLC}
}

@article{Akaike1969annals,
  title={Fitting autoregressive models for prediction},
  author={Akaike, Hirotugu},
  journal={Annals of the institute of Statistical Mathematics},
  volume={21},
  number={1},
  pages={243--247},
  doi={10.1007/BF02532251},
  year={1969},
  publisher={Springer}
}
@article{Brunton2017natcomm,
  title={Chaos as an intermittently forced linear system},
  author={Brunton, Steven L and Brunton, Bingni W and Proctor, Joshua L and Kaiser, Eurika and Kutz, J Nathan},
  journal={Nature communications},
  volume={8},
  number={1},
  pages={1--9},
  doi={10.1038/s41467-017-00030-8},
  year={2017},
  publisher={Nature Publishing Group}
}
@incollection{Nelles2013book,
  title={Nonlinear Dynamic System Identification},
  author={Nelles, Oliver},
  booktitle={Nonlinear System Identification},
  pages={831--891},
  year={2020},
  doi={10.1007/978-3-662-04323-3_15},
  publisher={Springer}
}
@article{rowley2009spectral,
  title={Spectral analysis of nonlinear flows},
  author={Rowley, Clarence W and Mezi{\'c}, Igor and Bagheri, Shervin and Schlatter, Philipp and Henningson, Dan S},
  journal={Journal of fluid mechanics},
  volume={641},
  pages={115--127},
  year={2009},
  doi={10.1017/S0022112009992059},
  publisher={Cambridge University Press}
}
@book{Billings2013book,
  title={Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains},
  author={Billings, Stephen A},
  year={2013},
  doi={10.1002/9781118535561},
  publisher={John Wiley \& Sons}
}
@inproceedings{long2017pde,
  title={Pde-net: Learning pdes from data},
  author={Long, Zichao and Lu, Yiping and Ma, Xianzhong and Dong, Bin},
  booktitle={International Conference on Machine Learning},
  pages={3208--3216},
  year={2018},
  doi={10.48550/arXiv.1710.09668},
  organization={PMLR}
}
@article{yang2020physics,
  title={Physics-informed generative adversarial networks for stochastic differential equations},
  author={Yang, Liu and Zhang, Dongkun and Karniadakis, George Em},
  journal={SIAM Journal on Scientific Computing},
  volume={42},
  number={1},
  pages={A292--A317},
  doi={10.1137/18M1225409},
  year={2020},
  publisher={SIAM}
}
@article{Wehmeyer2018jcp,
  title={Time-lagged autoencoders: Deep learning of slow collective variables for molecular kinetics},
  author={Wehmeyer, Christoph and No{\'e}, Frank},
  journal={The Journal of chemical physics},
  volume={148},
  number={24},
  pages={241703},
  year={2018},
  doi={10.1063/1.5011399},
  publisher={AIP Publishing LLC}
}
@article{Mardt2018natcomm,
  title={VAMPnets for deep learning of molecular kinetics},
  author={Mardt, Andreas and Pasquali, Luca and Wu, Hao and No{\'e}, Frank},
  journal={Nature communications},
  volume={9},
  number={1},
  pages={1--11},
  doi={10.1038/s41467-017-02388-1},
  year={2018},
  publisher={Nature Publishing Group}
}
@article{vlachas2018data,
  title={Data-driven forecasting of high-dimensional chaotic systems with long short-term memory networks},
  author={Vlachas, Pantelis R and Byeon, Wonmin and Wan, Zhong Y and Sapsis, Themistoklis P and Koumoutsakos, Petros},
  journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
  volume={474},
  number={2213},
  pages={20170844},
  doi={10.1098/rspa.2017.0844},
  year={2018},
  publisher={The Royal Society Publishing}
}
@article{pathak2018model,
  title={Model-free prediction of large spatiotemporally chaotic systems from data: A reservoir computing approach},
  author={Pathak, Jaideep and Hunt, Brian and Girvan, Michelle and Lu, Zhixin and Ott, Edward},
  journal={Physical review letters},
  volume={120},
  number={2},
  doi={10.1103/PhysRevLett.120.024102},
  pages={024102},
  year={2018},
  publisher={APS}
}

@article{lu2019deepxde,
  title={DeepXDE: A deep learning library for solving differential equations},
  author={Lu, Lu and Meng, Xuhui and Mao, Zhiping and Karniadakis, George Em},
  journal={SIAM Review},
  volume={63},
  number={1},
  pages={208--228},
  doi={10.1137/19M1274067},
  year={2021},
  publisher={SIAM}
}
@article{Raissi2019jcp,
title = {Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
journal = {Journal of Computational Physics},
volume = {378},
pages = {686-707},
year = {2019},
issn = {0021-9991},
doi = {10.1016/j.jcp.2018.10.045},
author = {Raissi, Maziar and Perdikaris, Paris and Karniadakis, George Em},
keywords = {Data-driven scientific computing, Machine learning, Predictive modeling, Runge–Kutta methods, Nonlinear dynamics},
}
@article{Champion2019pnas,
  title={Data-driven discovery of coordinates and governing equations},
  author={Champion, Kathleen and Lusch, Bethany and Kutz, J Nathan and Brunton, Steven L},
  journal={Proceedings of the National Academy of Sciences},
  volume={116},
  number={45},
  pages={22445--22451},
  year={2019},
  doi={10.1073/pnas.1906995116},
  publisher={National Acad Sciences}
}
@article{raissi2020science,
  title={Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations},
  author={Raissi, Maziar and Yazdani, Alireza and Karniadakis, George Em},
  journal={Science},
  volume={367},
  number={6481},
  pages={1026--1030},
  doi={10.1126/science.aaw4741},
  year={2020},
  publisher={American Association for the Advancement of Science}
}
@article{raissi2017parametric,
  title={Parametric Gaussian process regression for big data},
  author={Raissi, Maziar and Babaee, Hessam and Karniadakis, George Em},
  journal={Computational Mechanics},
  volume={64},
  pages={409--416},
  doi={10.1007/s00466-019-01711-5},
  year={2019},
  publisher={Springer}
}
@article{Benner2015siamreview,
  title={A survey of projection-based model reduction methods for parametric dynamical systems},
  author={Benner, Peter and Gugercin, Serkan and Willcox, Karen},
  journal={SIAM review},
  volume={57},
  number={4},
  pages={483--531},
  doi={10.1137/130932715},
  year={2015},
  publisher={SIAM}
}
@article{peherstorfer2016data,
  title={Data-driven operator inference for nonintrusive projection-based model reduction},
  author={Peherstorfer, Benjamin and Willcox, Karen},
  journal={Computer Methods in Applied Mechanics and Engineering},
  volume={306},
  pages={196--215},
  doi={10.1016/j.cma.2016.03.025},
  year={2016},
  publisher={Elsevier}
}

@article{qian2020lift,
  title={Lift \& learn: Physics-informed machine learning for large-scale nonlinear dynamical systems},
  author={Qian, Elizabeth and Kramer, Boris and Peherstorfer, Benjamin and Willcox, Karen},
  journal={Physica D: Nonlinear Phenomena},
  volume={406},
  pages={132401},
  doi={10.1016/j.physd.2020.132401},
  year={2020},
  publisher={Elsevier}
}

@article{Giannakis2012pnas,
  title={Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability},
  author={Giannakis, Dimitrios and Majda, Andrew J},
  journal={Proceedings of the National Academy of Sciences},
  volume={109},
  number={7},
  pages={2222--2227},
  doi={10.1073/pnas.1118984109},
  year={2012},
  publisher={National Acad Sciences}
}
@article{Yair2017pnas,
  title={Reconstruction of normal forms by learning informed observation geometries from data},
  author={Yair, Or and Talmon, Ronen and Coifman, Ronald R and Kevrekidis, Ioannis G},
  journal={Proceedings of the National Academy of Sciences},
  volume={114},
  number={38},
  pages={E7865--E7874},
  doi={10.1073/pnas.1620045114},
  year={2017},
  publisher={National Acad Sciences}
}

@article{bongard_automated_2007,
  title={Automated reverse engineering of nonlinear dynamical systems},
  author={Bongard, Josh and Lipson, Hod},
  journal={Proceedings of the National Academy of Sciences},
  volume={104},
  number={24},
  pages={9943--9948},
  doi={10.1073/pnas.0609476104},
  year={2007},
  publisher={National Acad Sciences}
}
@article{schmidt_distilling_2009,
  title={Distilling free-form natural laws from experimental data},
  author={Schmidt, Michael and Lipson, Hod},
  journal={science},
  volume={324},
  doi={10.1126/science.1165893},
  number={5923},
  pages={81--85},
  year={2009},
  publisher={American Association for the Advancement of Science}
}
@article{Daniels2015naturecomm,
  title={Automated adaptive inference of phenomenological dynamical models},
  author={Daniels, Bryan C and Nemenman, Ilya},
  journal={Nature communications},
  volume={6},
  number={1},
  pages={1--8},
  doi={10.1038/ncomms9133},
  year={2015},
  publisher={Nature Publishing Group}
}
@article{brunton2016pnas,
  title = {Discovering governing equations from data by sparse identification of nonlinear dynamical systems},
  author = {Brunton, Steven L. and Proctor, Joshua L. and Kutz, J. Nathan},
  journal = {Proceedings of the National Academy of Sciences},
  volume = {113},
  number = {15},
  pages = {3932--3937},
  year = {2016},
  publisher = {National Academy of Sciences},
  doi = {10.1073/pnas.1517384113}
}
@article{Williams2015jnls,
  title={A data--driven approximation of the {K}oopman operator: Extending dynamic mode decomposition},
  author={Williams, Matthew O and Kevrekidis, Ioannis G and Rowley, Clarence W},
  journal={Journal of Nonlinear Science},
  volume={25},
  number={6},
  pages={1307--1346},
  doi={10.1007/s00332-015-9258-5},
  year={2015},
  publisher={Springer}
}

@article{Rudy2017sciadv,
  Author = {Rudy, Samuel H and Brunton, Steven L and Proctor, Joshua L and Kutz, J Nathan},
  Journal = {Science Advances},
  Number = {e1602614},
  Title = {Data-driven discovery of partial differential equations},
  Volume = {3},
  Year = {2017},
  doi = {10.1126/sciadv.1602614}
}
@article{Williams2015jcd,
  title={A kernel approach to data-driven {K}oopman spectral analysis},
  author={Williams, Matthew O and Rowley, Clarence W and Kevrekidis, Ioannis G},
  journal={Journal of Computational Dynamics},
  volume={2},
  doi={10.3934/jcd.2015005},
  pages={247--265},
  year={2015}
}
@article{lusch2018deep,
  title={Deep learning for universal linear embeddings of nonlinear dynamics},
  author={Lusch, Bethany and Kutz, J Nathan and Brunton, Steven L},
  journal={Nature communications},
  volume={9},
  number={1},
  doi={10.1038/s41467-018-07210-0},
  pages={4950},
  year={2018},
  publisher={Nature Publishing Group UK London}
}
@article{otto2019linearly,
  title={Linearly recurrent autoencoder networks for learning dynamics},
  author={Otto, Samuel E and Rowley, Clarence W},
  journal={SIAM Journal on Applied Dynamical Systems},
  volume={18},
  number={1},
  doi={10.1137/18M1177846},
  pages={558--593},
  year={2019},
  publisher={SIAM}
}
@inproceedings{Takeishi2017nips,
	author = {Takeishi, Naoya and Kawahara, Yoshinobu and Yairi, Takehisa},
	booktitle = {Advances in Neural Information Processing Systems},
	pages = {1130--1140},
  doi={10.48550/arXiv.1710.04340},
	title = {Learning {K}oopman Invariant Subspaces for Dynamic Mode Decomposition},
	year = {2017}}

@article{pan2021sparsity,
  title={Sparsity-promoting algorithms for the discovery of informative {K}oopman-invariant subspaces},
  author={Pan, Shaowu and Arnold-Medabalimi, Nicholas and Duraisamy, Karthik},
  journal={Journal of Fluid Mechanics},
  volume={917},
  doi={10.1017/jfm.2021.271},
  pages={A18},
  year={2021},
  publisher={Cambridge University Press}
}
@article{surana2016linear,
  title={Linear observer synthesis for nonlinear systems using {K}oopman operator framework},
  author={Surana, Amit and Banaszuk, Andrzej},
  journal={IFAC-PapersOnLine},
  volume={49},
  number={18},
  pages={716--723},
  year={2016},
  doi={10.1016/j.ifacol.2016.10.250},
  publisher={Elsevier}
}

@article{korda2020optimal,
  title={Optimal construction of {K}oopman eigenfunctions for prediction and control},
  author={Korda, Milan and Mezi{\'c}, Igor},
  journal={IEEE Transactions on Automatic Control},
  volume={65},
  number={12},
  doi={10.1109/TAC.2020.2978039},
  pages={5114--5129},
  year={2020},
  publisher={IEEE}
}
@book{mauroy2020koopman,
  title={Koopman operator in systems and control},
  author={Mauroy, Alexandre and Susuki, Y and Mezi{\'c}, I},
  year={2020},
  doi={10.1007/978-3-030-35713-9},
  publisher={Springer}
}
@article{kaiser2021data,
  title={Data-driven discovery of {K}oopman eigenfunctions for control},
  author={Kaiser, Eurika and Kutz, J Nathan and Brunton, Steven L},
  journal={Machine Learning: Science and Technology},
  volume={2},
  number={3},
  doi={10.1088/2632-2153/abf0f5},
  pages={035023},
  year={2021},
  publisher={IOP Publishing}
}
@article{peitz2019koopman,
	author = {Peitz, Sebastian and Klus, Stefan},
	journal = {Automatica},
	pages = {184--191},
	title = {Koopman operator-based model reduction for switched-system control of PDEs},
	volume = {106},
  doi={10.1016/j.automatica.2019.05.016},
	year = {2019}}

@article{peitz2020data,
	author = {Peitz, Sebastian and Otto, Samuel E and Rowley, Clarence W},
	journal = {SIAM Journal on Applied Dynamical Systems},
	number = {3},
	pages = {2162--2193},
	title = {Data-driven model predictive control using interpolated {K}oopman generators},
	volume = {19},
  doi={10.1137/20M1325678},
	year = {2020}}

@article{de2020pysindy,
  doi = {10.21105/joss.02104},
  url = {https://doi.org/10.21105/joss.02104},
  year = {2020},
  publisher = {The Open Journal},
  volume = {5},
  number = {49},
  pages = {2104},
  author = {Brian M. de Silva and Kathleen Champion and Markus Quade and Jean-Christophe Loiseau and J. Nathan Kutz and Steven L. Brunton},
  title = {PySINDy: A {P}ython package for the sparse identification of nonlinear dynamical systems from data},
  journal = {Journal of Open Source Software}
}


@article{degennaro2019scalable,
  title={Scalable extended dynamic mode decomposition using random kernel approximation},
  author={DeGennaro, Anthony M and Urban, Nathan M},
  journal={SIAM Journal on Scientific Computing},
  volume={41},
  number={3},
  pages={A1482--A1499},
  year={2019},
  doi={10.1137/17M115414X},
  publisher={SIAM}
}

@article{mezic2004comparison,
  title={Comparison of systems with complex behavior},
  author={Mezi{\'c}, Igor and Banaszuk, Andrzej},
  journal={Physica D: Nonlinear Phenomena},
  volume={197},
  number={1-2},
  doi={10.1016/j.physd.2004.06.015},
  pages={101--133},
  year={2004},
  publisher={Elsevier}
}


@article{pan2020physics,
  title={Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability},
  author={Pan, Shaowu and Duraisamy, Karthik},
  journal={SIAM Journal on Applied Dynamical Systems},
  volume={19},
  number={1},
  doi={10.1137/19M1267246},
  pages={480--509},
  year={2020},
  publisher={SIAM}
}

@article{proctor2016dynamic,
  title={Dynamic mode decomposition with control},
  author={Proctor, Joshua L and Brunton, Steven L and Kutz, J Nathan},
  journal={SIAM Journal on Applied Dynamical Systems},
  volume={15},
  number={1},
  pages={142--161},
  year={2016},
  doi={10.1137/15M1013857},
  publisher={SIAM}
}


@article{hoffmann2021deeptime,
	author = {Hoffmann, Moritz and Scherer, Martin and Hempel, Tim and Mardt, Andreas and de Silva, Brian and Husic, Brooke E and Klus, Stefan and Wu, Hao and Kutz, Nathan and Brunton, Steven L and others},
	journal = {Machine Learning: Science and Technology},
	number = {1},
	pages = {015009},
	title = {Deeptime: a {P}ython library for machine learning dynamical models from time series data},
	volume = {3},
  doi={10.1088/2632-2153/ac3de0},
	year = {2021}}
